Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\text{Let I }=\int_\frac{1}{3}^1 \frac{\left( x - x^3 \right)^\frac{1}{3}}{x^4}dx\]
\[= \int_\frac{1}{3}^1 \frac{\left[ x^3 \left( \frac{x}{x^3} - 1 \right) \right]^\frac{1}{3}}{x^4}dx\]
\[ = \int_\frac{1}{3}^1 \frac{x \left( \frac{1}{x^2} - 1 \right)^\frac{1}{3}}{x^4}dx\]
\[ = \int_\frac{1}{3}^1 \frac{\left( \frac{1}{x^2} - 1 \right)^\frac{1}{3}}{x^3}dx\]
Put
\[\therefore - \frac{2}{x^3}dx = dz\]
\[ \Rightarrow \frac{dx}{x^3} = - \frac{dz}{2}\]
When
When
\[\therefore I = - \frac{1}{2} \int_8^0 z^\frac{1}{3} dz\]
\[ = \left.- \frac{1}{2} \times \frac{z^\frac{4}{3}}{\frac{4}{3}}\right|_8^0 \]
\[ = - \frac{3}{8}\left[ 0 - \left( 8 \right)^\frac{4}{3} \right]\]
\[ = - \frac{3}{8} \times \left( - 16 \right)\]
\[ = 6\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_0^{( \pi )^{2/3}} \sqrt{x} \cos^2 x^{3/2} dx\]
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
If f is an integrable function, show that
Prove that:
\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\] equals
The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is
The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is
Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .
\[\int\limits_0^4 x\sqrt{4 - x} dx\]
\[\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx\]
\[\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx\]
\[\int\limits_0^{2\pi} \cos^7 x dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx\]
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
\[\int\limits_0^\pi \frac{dx}{6 - \cos x}dx\]
\[\int\limits_2^3 e^{- x} dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 "e"^(2x) "d"x`
Evaluate the following integrals as the limit of the sum:
`int_1^3 x "d"x`
Choose the correct alternative:
If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x) "d"x + int_"c"^"b" f(x) "d"x` is
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
`int x^9/(4x^2 + 1)^6 "d"x` is equal to ______.
